(wave front sets of propagators of Klein-Gordon equation on Minkowski spacetime)
The wave front set of the various propagators for the Klein-Gordon equation on Minkowski spacetime, regarded, via translation invariance, as distributions in a single variable, are as follows:
the causal propagator $\Delta_S$ (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the lightcone:
the Hadamard propagator $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ on the light cone and $k^0 \gt 0$:
Regarding the causal propagator:
By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the singular support of $\Delta_S$ is the light cone.
Let $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ be a bump function whose compact support includes the origin.
For $a \in \mathbb{R}^{p,1}$ a point on the light cone, we need to determine the decay property of the Fourier transform of $x \mapsto b(x-a)\Delta_S(x)$. This is the convolution of distributions of $\hat b(k)e^{i k_\mu a^\mu}$ with $\widehat \Delta_S(k)$. By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have
This means that the convolution product is the smearing of the mass shell by $\widehat b(k)e^{i k_\u a^\mu}$.
Since the mass shell asymptotes to the light cone, and since $e^{i k_\mu a^\mu} = 1$ for $k$ on the light cone (given that $a$ is on the light cone), this implies immediately that all $k$ on the light cone are in the wave front set at the point $a$.
It remains to see that no other wave vctors $k$ are in the wave front set. But if $k$ is not on the light cone, the for large constants $c$ the product $c k$ has arbitrary large distance form the light cone. Since $\widehat b$ is a rapidly decreasing function, it fllows (..?..) that the convolution of the mass shell with $\widehat b$ is rapidly decreasing with distance from the light cone, hence rapidly decreasing along all $k$ not on the light cone.
Now for the Hadamard propagator:
By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form
First of all it follows that the singular support is still the light cone, because this means that $\Delta_H$ is a convolution of distributions of $\Delta_S$ with $\widehat {\Theta} \propto \delta'$, and this convolution does not increase the singular support (…).
Therefore now same argument as before says that the wave front set consists of wave vectors $k$ on the light cone, but now due to the step function factor $\Theta(-k_0)$ now they must be in only one of the two branches.